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I'm having trouble simplifying this expression:

$$\frac{1}{x} + \frac{5+x}{(x+1)} - \frac{7x^2 + 3}{(x+2)^2}$$

Would you first do the addition or subtraction?

What's the steps to solve this?

The final answer is

$$\frac{-6x^4 + 3x^3 + 26x^2 + 25x + 4}{x^4 + 5x^3 + 8x^2 + 4x}.$$

Thanks.

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2 Answers

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Or, if you set $I=\frac{1}x+\frac{x+5}{x+1}-\frac{7x^2+3}{(x+2)^2}$, then by multiplying $I$ by $x(x+1)(x+2)^2$, you get:

$$x(x+1)(x+2)^2\times I=x(x+1)(x+2)^2\left(\frac{1}x+\frac{x+5}{x+1}-\frac{7x^2+3}{(x+2)^2}\right)\\=(x+1)(x+2)^2+(x+5)x(x+2)^2-(7x^2+3)x(x+1)\\=3x^3+26x^2+25x+4-6x^4$$

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Well done, of course!+1 – amWhy Feb 8 at 0:04
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HINT

The guiding idea is the same as when you're evaluating $\frac{1}{3} + \frac{3}{4}$, which is to say that you find a common denominator. In my example, it would be $3\cdot 4$. In yours, it would be...

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Would it be x * (x + 1)? – Cypras Feb 7 at 8:03
That's a good guess - but that's only the common divisor for the first two fractions. – mixedmath♦ Feb 7 at 8:04
Yeah so I was wondering about that, is it better to do just two fractions at a time, or all three at once? – Cypras Feb 7 at 8:05
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@Cypras: The idea above rules the fractions properly. +1 – Babak S. Feb 7 at 8:13
Common denominator would be x? – Cypras Feb 7 at 8:15

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